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26                       ANALYTICAL MECHANICS
is equivalent to the given forces. There are two criteria by which this equivalence may be tested. First: The resultant force will give the particle the same motion, when applied to it, as that imparted by the given system of forces. We cannot use this test just now because we have not yet •studied motion. Second: When the resultant force is reversed and applied to the particle simultaneously with the given forces the particle remains in equilibrium.
According to the second criterion, therefore, the resultant, R, of the forces Fi, F2, . . . , F», must satisfy the equation
-R+(Fx+F2 + - • • +Fn) = or                    R = F!+F2+- • - +Fn.
Splitting the last equation into three algebraic equations, we obtain
-sr =
Z = Z1+Z2 + - • • +Zn,
where Xi} Yt, and Zt are the components of F^. The magnitude of R is given by the relation
R = VlX*~+Y*+Z~,                          (v')
while the direction is obtained from the following expressions for its direction cosines.
X                Y                 7
COS a\ = —>    COS &2 = •£•;    COS «3 =    ' •              ( VI/)
**                      &                       It
_    Special Case.— Wlxen the forces lie in the rr/y-plune the c2-component of each force equals zero.   Thc>r<;fom wo have
(V) (
e is the angle R makes with the re-axis.